In problems without contexts, the properties we highlight are largely known in the literature. They follow from analyses of top-two sampling algorithms initiated by [1]} and extended in conference papers of [2]} and [3]} as well as lower bound techniques of [4]} and [5]}.
Our work extends these properties to address contextual problems. Because DTS sampling probabilities are context-independent, as highlighted in Section , extending its asymptotic analysis to contextual case is technical demanding, but uses similar ideas to past work. The main new insight is in the lower bound, where we need to show that context-independent sampling is optimal. This means that an algorithm that observed contexts after choosing an arm can perform as well asymptotically as one which observed contexts prior to arm selection.
The key step there is in Proposition REF  in the appendix.
